Problem: Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
Answer: Because the 6 empty chairs are $\frac{1}{4}$ of the chairs in the room, there are $6\times 4=24$ chairs in all. The number of seated people is $\frac{3}{4}\times24=18$, and this is $\frac{2}{3}$ of the people present. It follows that \[\frac{18}{\textrm{people present}}=\frac{2}{3}.\]So there are $\boxed{27}$ people in the room.